Initial investigation into the complementary use of black box and physics-based techniques in rotorcraft system identification

Abstract

Accurate linear helicopter models are needed for control system development and simulation and can be determined by system identification when appropriate test data are available. Standard methods for rotorcraft system identification are the frequency domain maximum likelihood method and the frequency response method that are used to derive physics-based linear state-space models. Also the optimized predictor-based subspace identification method (PBSIDopt), a time domain system identification method that yields linear black box state-space models, has been successfully applied to rotorcraft data. As both methods have their respective strengths and weaknesses, it was tried to combine both techniques. The paper demonstrates the successful complementary use of physics-based frequency domain methods and the black box PBSIDopt method in the areas of database requirements, accuracy metrics, and model structure development using flight test data of DLR’s ACT/FHS research rotorcraft.

Appendix: Applied identification methods

$\varvec{A}_d$, $\varvec{B}_d$, $\varvec{C}_d$, $\varvec{D}_d$:

Discrete-time state-space matrices

$\varvec{A}_K$, $\varvec{B}_K$:

Predictor form state-space matrices

$\varvec{e}_k$, $\varvec{u}_k$, $\varvec{x}_k$, $\varvec{y}_k$:

Discrete-time innovation, input, state, and output vectors at kth time step

$\varvec{E}$, $\varvec{U}$, $\varvec{X}$, $\varvec{Y}$:

Data matrices for system innovation, input, state, and output

f, p:

Future and past window length

$\varvec{K}$ :

Kalman gain matrix

$\varvec{\mathcal {K}}^{(p)}$ :

Extended controllability matrix

$M_f$, $M_n$, $M_p$:

Sets for f, n and p

$n_u$ :

Number of model inputs

N :

Number of data points

$\varvec{\mathcal {O}}^{(f)}$ :

Extended observability matrix

s :

Laplace variable (1/s)

$\varvec{\mathcal {S}}$ :

Diagonal singular values matrix

$\varvec{T}$ :

Frequency response matrix

$w_{\mathrm{{ap}}}$ :

Relative weighting amplitude/phase errors

$w_{\gamma }$ :

Coherence weighting

$\varvec{y}_{\mathrm{{m}},k}$ :

Measured output (index m)

$\varvec{z}_k$ :

Merged input–output vector at kth time step

$\varvec{Z}$ :

Data matrices for merged input–outputs (used with indexes)

$\gamma _{uy}^2$ :

Coherence between u and y

$\lambda$ :

Regularization parameter

$\omega$ :

Angular frequency (rad/s)

$\sigma (\ldots )$ :

Standard deviation

$\tau$ :

Time delay (s)

$\measuredangle$ :

Phase angle ($^{\circ }$)

$|\ldots |_{\text{dB}}$ :

Amplitude (dB)

1.1 ML output error method

The system to be identified is assumed to be described by a linear state-space model

$$\begin{aligned} \begin{aligned} \dot{\varvec{x}}(t)&= \varvec{A}(\varvec{\varTheta }) \varvec{x}(t) + \varvec{B}(\varvec{\varTheta }) \varvec{u}(t)\\ \varvec{y} (t)&= \varvec{C}(\varvec{\varTheta }) \varvec{x}(t) + \varvec{D}(\varvec{\varTheta }) \varvec{u}(t), \end{aligned} \end{aligned}$$

(6)

where $\varvec{x}$ denotes the state vector, $\varvec{u}$ the input vector and $\varvec{y}$ the output vector. The system matrices $\varvec{A}$, $\varvec{B}$, $\varvec{C}$ and $\varvec{D}$ contain the unknown model parameters $\varvec{\varTheta }$. Measurements $\varvec{z}$ of the outputs exist for N discrete time points $t_k$

$$\begin{aligned} \varvec{z}_k = \varvec{y}(t_k) + \varvec{v}(t_k) \ , \ \ \ k=1, \dots , N. \end{aligned}$$

(7)

The measurement noise $\varvec{v}$ is assumed to be characterized by Gaussian white noise with covariance matrix $\varvec{R}$.

The ML estimates of the unknown parameters $\varvec{\varTheta }$ and of the measurement noise covariance matrix $\varvec{R}$ are obtained by minimizing the cost function

$$\begin{aligned} \begin{aligned} J(\varvec{\varTheta },\varvec{R})&= \frac{1}{2} \sum _{k=1}^{N}{\left[ \varvec{z}(t_k)-\varvec{y}(t_k) \right] ^T \varvec{R}^{-1}} \\&\times \left[ \varvec{z}(t_k)-\varvec{y}(t_k) \right] + \frac{N}{2} \ln {\left( \det {(\varvec{R})}\right) } . \end{aligned} \end{aligned}$$

(8)

If the measurement error covariance matrix $\varvec{R}$ is unknown, as it is usually the case, the optimization of Eq. (8) is carried out in two steps. In the first step, it can be shown that for any given value of $\varvec{\varTheta }$, the ML estimate of $\varvec{R}$ is given by

$$\begin{aligned} \varvec{R} = \frac{1}{N} \sum _{k=1}^{N}{\left[ \varvec{z}(t_k)-\varvec{y}(t_k) \right] \left[ \varvec{z}(t_k)-\varvec{y}(t_k) \right] ^T } \end{aligned}$$

(9)

which means that the output error covariance matrix is the most plausible estimate for $\varvec{R}$.

Thus, the variable part of the cost function reduces to

$$\begin{aligned} J(\varvec{\varTheta }) = \ln {\left( \det {(\varvec{R})}\right) }. \end{aligned}$$

(10)

If the covariance matrix $\varvec{R}$ is assumed to be a diagonal matrix, the cost function reduces to the product of the output error variances of all output variables

$$\begin{aligned} J(\varvec{\varTheta }) = \prod _{j=1}^{n_y} \left( \frac{1}{N} \sum _{k=1}^{N}{\left[ \varvec{z}_j(t_k)-\varvec{y}_j(t_k) \right] ^2} \right) . \end{aligned}$$

(11)

Frequency domain variant

The discretely sampled time-dependent variable

$$\begin{aligned} \varvec{x}_k = \varvec{x}(k\varDelta t) \ , \ \ \ k=0, \ldots , N-1 \end{aligned}$$

(12)

with the sampling time interval $\varDelta t$ is transformed to a frequency-dependent variable using the Fourier transform

$$\begin{aligned} \begin{aligned} \varvec{x}(\omega _k)&= \frac{1}{N} \sum _{k=0}^{N-1} \varvec{x}_k \text {e}^{-\text {i}\omega _k k \varDelta t} \\ \omega _k&=k \cdot 2\pi / t_N \ \ \text {with} \ \ t_N =(N-1)\varDelta t \ . \end{aligned} \end{aligned}$$

(13)

Transforming the variables $\dot{\varvec{x}}$, $\varvec{x}$, $\varvec{u}$, $\varvec{y}$ of the linear model from Eq. (6) to the frequency domain leads to the following model equations in the frequency domain

$$\begin{aligned} \begin{aligned} \text {i} \omega \varvec{x}(\omega )&= \varvec{A}(\varvec{\varTheta }) \varvec{x}(\omega ) + \varvec{B}(\varvec{\varTheta }) \varvec{u}(\omega ) \\ \varvec{y}(\omega )&= \varvec{C}(\varvec{\varTheta }) \varvec{x}(\omega ) + \varvec{D}(\varvec{\varTheta }) \varvec{u}(\omega ). \end{aligned} \end{aligned}$$

(14)

The ML cost function in the frequency domain is derived analogously to the one in the time domain with the output error covariance matrix $\varvec{R}$ replaced by the spectral density matrix of the measurement noise. The ML cost function in the frequency domain is, therefore,

$$\begin{aligned} J(\varvec{\varTheta }) = \prod _{i=j}^{n_y} \varvec{\sigma }^2 \left( \varvec{z}_j - \varvec{y}_j \right) \end{aligned}$$

(15)

with

$$\begin{aligned} \begin{aligned}&\varvec{\sigma }^2 \left( \varvec{z}_j - \varvec{y}_j \right) = \frac{1}{N}\sum _{k=0}^{N-1} {\left[ \varvec{z}_j(\omega _k)-\varvec{y}_j(\omega _k) \right] ^*\left[ \varvec{z}_j(\omega _k)-\varvec{y}_j(\omega _k) \right] }, \end{aligned} \end{aligned}$$

(16)

where $(.)^*$ denotes the conjugate transpose of a complex value and $\varvec{\sigma }^2 (.)$ the model error variance.

Minimization of the cost function from Eq. (11) or Eq. (15) is performed using, e.g., a Gauss–Newton optimization method.

1.2 Frequency response method

The ML method in the frequency domain is based on matching the Fourier transform of the output variables. In contrast, the frequency response method is based on matching the frequency responses, i.e., the ratio of the output per unit of control input as a function of control input frequency.

The frequency response matrix of the identification model $\varvec{T}(s)$ relates the Laplace transform $\varvec{Y}(s)$ of the output vector $\varvec{y}$ to the Laplace transform $\varvec{U}(s)$ of the input vector $\varvec{u}$:

$$\begin{aligned} \varvec{Y}(s) = \varvec{T}(s) \varvec{U}(s). \end{aligned}$$

(17)

For the linear state-space system from Eq. (6), the frequency response matrix is determined as

$$\begin{aligned} \varvec{T}(s) = \varvec{C}(s\varvec{I}-\varvec{A})^{-1}\varvec{B} + \varvec{D}, \end{aligned}$$

(18)

where $\varvec{I}$ denotes the identity matrix.

The quadratic cost function to be minimized for the frequency response method is

$$\begin{aligned} \begin{aligned} J = \frac{20}{N_\omega }&\sum _{k=1}^{N\omega }{ w_\gamma (k) \left[ \left( |T_\mathrm {m}(k)|_{dB}-|T(k)|_{dB}\right) ^2 \right. } \\ &\left. + \; w_{ap} \left( \measuredangle T_\mathrm {m}(k)-\measuredangle T(k) \right) ^2\right] , \end{aligned} \end{aligned}$$

(19)

where T and $T_\mathrm{{m}}$ are a single-frequency response and its measured counterpart. $N_\omega$ is the number of frequency points in the frequency interval $[\omega _1,\omega _{N\omega }]$. $|\ldots |_{dB}$ denotes the amplitude in dB and $\measuredangle (\ldots )$ the phase angle in degree.

$w_\gamma$ is an optional weighting function based on the coherence between the input and the output at each frequency. It is defined as

$$\begin{aligned} w_\gamma (k) = \left[ 1.58 (1-e^{\gamma ^2_{xy}(k)}) \right] ^2, \end{aligned}$$

(20)

$w_{\mathrm{{ap}}}$ is the relative weight between amplitude and phase errors. The normal convention is $w_{\mathrm{{ap}}} = 0.01745$.

When several frequency responses are approximated together, the overall cost function is the average of the individual cost functions. A good overview of system identification using the frequency response method can be found in Ref. [2].

1.3 PBSIDopt method

The starting point for the PBSIDopt method is a linear discrete-time state-space model in innovation form

$$\begin{aligned} \begin{aligned} \varvec{x}_{k+1}&= \varvec{A}_d \varvec{x}_k + \varvec{B}_d \varvec{u}_k + \varvec{K} \varvec{e}_k \\ \varvec{y}_k&= \varvec{C}_d \varvec{x}_k + \varvec{D}_d \varvec{u}_k + \varvec{e}_k \end{aligned} \end{aligned}$$

(21)

with the input vector $\varvec{u}_k \in {\mathbb {R}}^{n_u}$, the outputs $\varvec{y}_k \in {\mathbb {R}}^{n_y}$ and the states $\varvec{x}_k \in {\mathbb {R}}^{n}$. The innovations $\varvec{e}_k \in {\mathbb {R}}^{n_y}$ are assumed to be zero-mean white process noise. A finite set of data points $\varvec{u}_k$ and $\varvec{y}_k$ with $k = 1\; \ldots N$ is considered for system identification.

Assuming there is no direct feedthrough, i.e., $\varvec{D}_\mathrm{{d}}=\varvec{0}$, the system in Eq. (21) is transformed to the one-step-ahead predictor form

$$\begin{aligned} \begin{aligned} \varvec{x}_{k+1}&= \varvec{A}_K \varvec{x}_k + \varvec{B}_K \varvec{z}_k \\ \varvec{y}_k&= \varvec{C}_\mathrm{{d}} \varvec{x}_k + \varvec{e}_k \end{aligned} \end{aligned}$$

(22)

with $\varvec{A}_K = \varvec{A}_\mathrm{{d}} - \varvec{K} \varvec{C}_\mathrm{{d}}$, $\varvec{B}_K = \left( \varvec{B}_\mathrm{{d}} \ \ \varvec{K} \right)$ and $\varvec{z}_k = (\varvec{u}_k \ \ \varvec{y}_k )^T$. Furthermore, it is assumed that all eigenvalues of $\varvec{A}_K$ are inside the unit circle. Accordingly, the given predictor model is stable. The ($k+p$)th state $\varvec{x}_{k+p}$ is given by

$$\begin{aligned} \begin{aligned}&\varvec{x}_{k+p} = \varvec{A}_K \varvec{x}_{k+p-1} + \varvec{B}_K \varvec{z}_{k+p-1} \\&= \varvec{A}_K^p \varvec{x}_k + \underbrace{\begin{pmatrix} \varvec{A}_K^{p-1} \varvec{B}_K&\varvec{A}_K^{p-2} \varvec{B}_K&\ldots&\varvec{B}_K \end{pmatrix} }_{\varvec{\mathcal {K}}^{(p)}} \begin{pmatrix} \varvec{z}_k \\ \varvec{z}_{k+1} \\ \vdots \\ \varvec{z}_{k+p-1} \end{pmatrix} \end{aligned} \end{aligned}$$

(23)

and the ($k+p$)th output $\varvec{y}_{k+p}$ is determined

$$\begin{aligned} \varvec{y}_{k+p} = \varvec{C}_\mathrm{{d}} \varvec{A}_K^p \varvec{x}_k + \varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)} \begin{pmatrix} \varvec{z}_k \\ \varvec{z}_{k+1} \\ \vdots \\ \varvec{z}_{k+p-1} \end{pmatrix} + \varvec{e}_{k+p} \end{aligned}$$

(24)

with the extended controllability matrix $\varvec{\mathcal {K}}^{(p)}$ and the past window length p. Since $\varvec{A}_K$ is stable, the expression $\varvec{A}_K^p$ in Eqs. (23) and  (24) can be neglected for large p: $\varvec{A}_K^p \simeq \varvec{0}$. Therefore, repeating Eqs. (23) and (24) for the (p+1)th in the Nth element yields

$$\begin{aligned} \varvec{X}_{(p+1,N)}&= \varvec{\mathcal {K}}^{(p)} \varvec{Z}_{(1,N-p),p} \end{aligned}$$

(25a)

$$\begin{aligned} \varvec{Y}_{(p+1,N)}&= \varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)} \varvec{Z}_{(1,N-p),p} + \varvec{E}_{(p+1,N)} \end{aligned}$$

(25b)

with

$$\begin{aligned} \varvec{X}_{(p+1,N)} = \begin{pmatrix} \varvec{x}_{p+1}&\varvec{x}_{p+2}&\ldots&\varvec{x}_{N} \end{pmatrix} \end{aligned}$$

(26)

and analogous definitions for $\varvec{Y}_{(p+1,N)}$ and $\varvec{E}_{(p+1,N)}$. The merged inputs and outputs are combined as

$$\begin{aligned} \varvec{Z}_{(1,N-p),p} = \begin{pmatrix} \varvec{z}_{1} &{} \varvec{z}_{2} &{} \ldots &{} \varvec{z}_{N-p} \\ \varvec{z}_{2} &{} \varvec{z}_{3} &{} \ldots &{} \varvec{z}_{N-p+1} \\ \vdots &{} \vdots &{} \ldots &{} \vdots \\ \varvec{z}_{p} &{} \varvec{z}_{p+1} &{} \ldots &{} \varvec{z}_{N-1} \end{pmatrix}. \end{aligned}$$

(27)

The predictor Markov parameters $\varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)}$ are estimated in a least-squares sense with Tikhonov regularization to prevent ill-posed problems. The regularized least-squares problem is given by

$$\begin{aligned} \begin{aligned} \min _{\varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)}}&\left( \left\| \varvec{Y}_{(p+1,N)} - \varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)} \varvec{Z}_{(1,N-p),p} \right\| _F^2 \right. + \left. \lambda ^2 \left\| \varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)} \right\| _F^2 \right) . \end{aligned} \end{aligned}$$

(28)

The regularization parameter $\lambda$ is chosen with the strong robust generalized cross-validation method, see Ref. [23] for an introduction and a comparison of parameter choice methods.

The estimated predictor Markov parameters $\varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)}$ can be interpreted as a high-order vector-ARX model (AutoRegressive model with eXogenous input). High-order ARX models based on Eq. (25b) are asymptotically unbiased by correlation issues for large N and p, see Ref. [24]. Thus, this step is essential for subspace identification methods such as PBSIDopt to provide consistent estimates even in correlated closed-loop experiments.

Defining the extended observability matrix $\varvec{\mathcal {O}}^{(f)}$ with the future window length f

$$\begin{aligned} \varvec{\mathcal {O}}^{(f)} = \begin{pmatrix} \varvec{C}_\mathrm{{d}} \\ \varvec{C}_\mathrm{{d}} \varvec{A}_K \\ \vdots \\ \varvec{C}_\mathrm{{d}} \varvec{A}_K^{f-1} \end{pmatrix}, \end{aligned}$$

(29)

the product of the extended observability matrix $\varvec{\mathcal {O}}^{(f)}$ and the extended controllability matrix $\varvec{\mathcal {K}}^{(p)}$ is set up using the estimated predictor Markov parameters $\varvec{C}_\mathrm{{d}} \varvec{\mathcal {K}}^{(p)}$

$$\begin{aligned} \begin{aligned}&\varvec{\mathcal {O}}^{(f)} \varvec{\mathcal {K}}^{(p)} \simeq \begin{pmatrix} \varvec{C}_\mathrm{{d}} \varvec{A}_K^{p-1} \varvec{B}_K &{} \varvec{C}_\mathrm{{d}} \varvec{A}_K^{p-2} \varvec{B}_K &{} \ldots &{} \varvec{C}_\mathrm{{d}} \varvec{B}_K \\ \varvec{0} &{} \varvec{C}_\mathrm{{d}} \varvec{A}_K^{p-1} \varvec{B}_K &{} \ldots &{} \varvec{C}_\mathrm{{d}} \varvec{A}_K \varvec{B}_K \\ \vdots &{} \ddots &{} \ddots &{} \vdots \\ \varvec{0} &{} &{} &{} \varvec{C}_\mathrm{{d}} \varvec{A}_K^{f-1} \varvec{B}_K. \end{pmatrix} \end{aligned} \end{aligned}$$

(30)

According to Eq. (25a)

$$\begin{aligned} \begin{aligned} \varvec{\mathcal {O}}^{(f)} \varvec{X}_{(p+1,N)}&= \varvec{\mathcal {O}}^{(f)} \varvec{\mathcal {K}}^{(p)} \varvec{Z}_{(1,N-p),p} \\&= \varvec{\mathcal {U}} \varvec{\mathcal {S}} \varvec{\mathcal {V}}^T \\&= \begin{pmatrix} \varvec{\mathcal {U}}_n&\varvec{\mathcal {U}}_{{\overline{n}}} \end{pmatrix} \begin{pmatrix} \varvec{\mathcal {S}}_n &{} \varvec{0} \\ \varvec{0} &{} \varvec{\mathcal {S}}_{{\overline{n}}} \end{pmatrix} \begin{pmatrix} \varvec{\mathcal {V}}^T_n \\ \varvec{\mathcal {V}}^T_{{\overline{n}}}, \end{pmatrix} \end{aligned} \end{aligned}$$

(31)

the singular value decomposition is applied to reconstruct an estimation of the system states

$$\begin{aligned} \widetilde{\varvec{X}}_{(p+1,N)} = \varvec{\mathcal {S}}_n^{\frac{1}{2}} \varvec{\mathcal {V}}_n^T. \end{aligned}$$

(32)

The model order n corresponds to the n largest singular values in $\varvec{\mathcal {S}}_n$ used for the state sequence reconstruction.

Finally, the system matrices $\varvec{A}_\mathrm{{d}}$, $\varvec{B}_\mathrm{{d}}$, $\varvec{C}_\mathrm{{d}}$ and $\varvec{K}$ from Eq. (21) are calculated. First,

$$\begin{aligned} \begin{pmatrix} \widetilde{\varvec{X}}_{(\mathtt {p}+2,N)} \\ \varvec{Y}_{(\mathtt {p}+1,N-1)} \end{pmatrix} = \begin{pmatrix} \varvec{A}_\mathrm{{d}} &{} \varvec{B}_\mathrm{{d}} \\ \varvec{C}_\mathrm{{d}} &{} \varvec{0} \end{pmatrix} \begin{pmatrix} \widetilde{\varvec{X}}_{(\mathtt {p}+1,N-1)} \\ \varvec{U}_{(\mathtt {p}+1,N-1)} \end{pmatrix} \end{aligned}$$

(33)

is solved for $\varvec{A}_\mathrm{{d}}$, $\varvec{B}_\mathrm{{d}}$ and $\varvec{C}_\mathrm{{d}}$ in a least-squares sense. The Kalman gain $\varvec{K}$ is then calculated from the covariance matrix of the least-squares residuals and the system matrices $\varvec{A}_\mathrm{{d}}$ and $\varvec{C}_\mathrm{{d}}$ by solving the stabilizing solution of the corresponding discrete-time algebraic Riccati equation, see Ref. [24] and the references therein.

The inverse bilinear (or any other discrete time to continuous time) transform is then applied to calculate the continuous-time state-space model:

$$\begin{aligned} \begin{aligned} \dot{\varvec{x}}(t)&= \varvec{A} \varvec{x}(t) + \varvec{B} \varvec{u}(t) \\ \varvec{y}(t)&= \varvec{C} \varvec{x}(t) . \end{aligned} \end{aligned}$$

(34)

Selecting only the largest n singular values to reconstruct the state sequence in Eq. (32) already corresponds to a model reduction step. If necessary, further model reduction techniques as described in Ref. [12] can be used to adapt a high-order black box model to the frequency range of interest or to reduce its complexity. In the examples presented in this paper, no further model reduction techniques were applied.